q-Schur Algebras as Quotients of Quantized Enveloping Algebras

Title: Q-schur Algebras as Quotients of Quantized Enveloping Algebras

Constructing Canonical Bases of Quantized Enveloping Algebras

Since the invention of canonical bases of quantized enveloping algebras, one of the main problems has been to establish what they look like. Explicit formulas are only known in a few cases corresponding to root systems of low rank, namely A1 (trivial), A2 ([Lusztig 90]), A3 ([Xi 99a]), and B2 ([Xi 99b]). Furthermore, there is evidence suggesting that for higher ranks the formulas become so comp...

Cohomological construction of quantized universal enveloping algebras

Given an associative algebra A, and the category, C, of its finite dimensional modules, additional structures on the algebra A induce corresponding ones on the category C. Thus, the structure of a rigid quasi-tensor (braided monoidal) category on RepA is induced by an algebra homomorphism A → A ⊗ A (comultiplication), coassociative up to conjugation by Φ ∈ A (associativity constraint) and cocom...

Monomial Bases of Quantized Enveloping Algebras

We construct a monomial basis of the positive part U of the quantized enveloping algebra associated to a finite–dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of U. 0. Introduction In [L1], Lusztig showed that the positive part U of the quantized enveloping algebra associated to a finite–dimensional simple Lie alge...

GRADED q-SCHUR ALGEBRAS

Generalizing recent work of Brundan and Kleshchev, we introduce grading on Dipper-James’ q-Schur algebra, and prove a graded analogue of the Leclerc and Thibon’s conjecture on the decomposition numbers of the q-Schur algebra.